ellipse arc length

(0.001 seconds)

11—20 of 35 matching pages

11: 31.11 Expansions in Series of Hypergeometric Functions

… ►and (31.11.1) converges to (31.3.10) outside the ellipse

\mathcal{E}

in the

z

-plane with foci at 0, 1, and passing through the third finite singularity at

z=a

. … ►The expansion (31.11.1) for a Heun function that is associated with any branch of (31.11.2)—other than a multiple of the right-hand side of (31.11.12)—is convergent inside the ellipse

\mathcal{E}

. … ►For Heun functions (§31.4) they are convergent inside the ellipse

\mathcal{E}

. …

12: 4.15 Graphics

… ►Lines parallel to the real axis in the

z

-plane map onto ellipses in the

w

-plane with foci at

w=\pm 1

, and lines parallel to the imaginary axis in the

z

-plane map onto rectangular hyperbolas confocal with the ellipses. …

13: 32.14 Combinatorics

… ►With

1\leq m_{1}<\cdots<m_{n}\leq N

\boldsymbol{\pi}(m_{1}),\boldsymbol{\pi}(m_{2}),\dots,\boldsymbol{\pi}(m_{n})

is said to be an increasing subsequence of

\boldsymbol{\pi}

of length

n

when

\boldsymbol{\pi}(m_{1})<\boldsymbol{\pi}(m_{2})<\cdots<\boldsymbol{\pi}(m_{n})

. Let

\ell_{N}(\boldsymbol{\pi})

be the length of the longest increasing subsequence of

\boldsymbol{\pi}

. … ►

32.14.1

\lim_{N\to\infty}\mathrm{Prob}\left(\frac{\ell_{N}(\boldsymbol{\pi})-2\sqrt{N}% }{N^{1/6}}\leq s\right)=F(s),

ⓘ

Symbols:: $\boldsymbol{\pi}(m)$ : permutations, $\ell_{N}(\boldsymbol{\pi})$ : length and $F(s)$ : distribution function
Permalink:: http://dlmf.nist.gov/32.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §32.14 and Ch.32

…

14: 14.24 Analytic Continuation

… ►Let

s

be an arbitrary integer, and

P^{-\mu}_{\nu}\left(ze^{s\pi i}\right)

and

\boldsymbol{Q}^{\mu}_{\nu}\left(ze^{s\pi i}\right)

denote the branches obtained from the principal branches by making

\frac{1}{2}s

circuits, in the positive sense, of the ellipse having

\pm 1

as foci and passing through

z

. …

15: 23.23 Tables

… ►The values are tabulated on the real and imaginary

z

-axes, mostly ranging from 0 to 1 or

i

in steps of length 0. …

16: 26.13 Permutations: Cycle Notation

… ►Cycles of length one are fixed points. … ►An element of

\mathfrak{S}_{n}

with

a_{1}

fixed points,

a_{2}

cycles of length

2,\ldots,a_{n}

cycles of length

n

, where

n=a_{1}+2a_{2}+\cdots+na_{n}

, is said to have cycle type

{\left(a_{1},a_{2},\ldots,a_{n}\right)}

. … ►A transposition is a permutation that consists of a single cycle of length two. …A permutation that consists of a single cycle of length

k

can be written as the composition of

k-1

two-cycles (read from right to left): …

17: 19.33 Triaxial Ellipsoids

… ►For additional geometrical properties of ellipsoids (and ellipses), see Carlson (1964, p. 417). …

18: 28.32 Mathematical Applications

… ►If the boundary conditions in a physical problem relate to the perimeter of an ellipse, then elliptical coordinates are convenient. …

19: 19.9 Inequalities

… ►The perimeter

L(a,b)

of an ellipse with semiaxes

a, b

is given by …Even for the extremely eccentric ellipse with

a=99

and

b=1

, this is correct within 0. …

20: 4.1 Special Notation

… ►Sometimes “arc” is replaced by the index “

-1

”, e. …